On-shell diagrams and the geometry of planar N < 4 SYM theories

TL;DR

This paper explores the geometric structure of decorated on-shell diagrams in planar N<4 SYM theories, revealing new higher-order singularities and their relation to Grassmannian varieties, with implications for understanding scattering amplitudes.

Contribution

It introduces a novel decoration of on-shell diagrams that tracks helicity, leading to new on-shell functions and higher-order singularities in the Grassmannian framework.

Findings

Identification of higher-codimension varieties via Plucker coordinate relations

Introduction of distributional support on derivative delta-functions

Derivation of identities among on-shell functions using residue theorem

Abstract

We continue the discussion of the decorated on-shell diagrammatics for planar N < 4 Supersymmetric Yang-Mills theories started in arXiv:1510.03642. In particular, we focus on its relation with the structure of varieties on the Grassmannian. The decoration of the on-shell diagrams, which physically keeps tracks of the helicity of the coherent states propagating along their edges, defines new on-shell functions on the Grassmannian and can introduce novel higher-order singularities, which graphically are reflected into the presence of helicity loops in the diagrams. These new structures turn out to have similar features as in the non-planar case: the related higher-codimension varieties are identified by either the vanishing of one (or more) Plucker coordinates involving at least two non-adjacent columns, or new relations among Plucker coordinates. A distinctive feature is that the functions living on these higher-codimenson varieties can be thought of distributionally as having support on derivative delta-functions. After a general discussion, we explore in some detail the structures of the on-shell functions on Gr(2,4) and Gr(3,6) on which the residue theorem allows to obtain a plethora of identities among them.